Mathematics: ABEGG'S RULE, ABEL'S THEOREM, ARCHIMEDES' PROBLEM, BERNOULLI'S THEOREM, DE MOIVRE'S THEOREM, DE MORGAN'S THEOREM, DESARGUES' THEOREM,

DESCARTES' RULE OF SIGNS, EUCLID'S ALGORITHM, EULER'S EQUATION/FORMULA, FERMAT'S PRINCIPLE, FOURIER'S THEOREM, GAUSS'S THEOREM, GOLDBACH'S CONJECTURE, HUDDE'S RULES, LAPLACE'S EQUATIONS, NEWTON'S METHOD/PARALLELOGRAM, PASCAL'S LAW/TRIANGLE, RIEMANN'S HYPOTHESIS

Using a combination of Descartes' rule of signs and various geometric arguments for the roots of the equations, we proved the following theorem:

In the polynomial [p.sub.1](x) = [[mu].sub.1][[mu].sub.2.sup.2][x.sup.4] - 2[K.sub.2][[mu].sub.1][[mu].sub.2][x.sup.2] + x + [[mu].sub.1][K.sub.2.sup.2] - [K.sub.1]' (which is defined by the left hand side of equation 1.5) there are sign changes among the coefficients [[mu].sub.1][[mu].sub.2.sup.2], -2[K.sub.2][[mu].sub.1][[mu].sub.2], 1 and ([[mu].sub.1][K.sub.2.sup.2] - [K.sub.1]); therefore, by Descartes' rule of signs, there are either three positive zeroes or there is one positive zero for the polynomial.

Using Descartes' rule of signs for the roots of equation (1.8), we can conclude that the sign of 1 - 4[[mu].sub.1][[mu].sub.2][x.sub.c], [y.sub.c] determines the sign of the eigenvalues.

More specifically, the paper demonstrates the applicability of Descartes' Rule of Signs, Budan's Theorem, and Sturm's Theorem from the theory of equations and rules developed in the business literature by Teichroew, Robichek, and Montalbano (1965a, 1965b), Mao (1969), Jean (1968, 1969), and Pratt and Hammond (1979).

Of particular interest are Descartes' Rule of Signs, Budan's Theorem, and Sturm's Theorem from the theory of equations, and the rules developed in the business literature by Teichroew, Robichek, and Montalbano (1965a, 1965b), Mao (1969), Jean (1968, 1969), and Pratt and Hammond (1979).

In addition, Descartes' Rule of Signs is actually a special case of Budan's Theorem, which will be discussed next.

By, in turn, using Descartes' Rule of Signs, Budan's Theorem, and Sturm's Theorem, an increasingly clearer picture is gained of the number of positive solutions to the pump problem.